Highest weight modules with respect to non-standard Gelfand-Tsetlin subalgebras

TL;DR

This paper explores how highest weight modules for the complex Lie algebra gl_n can be realized using non-standard Gelfand-Tsetlin subalgebras, including conditions for diagonalizability of their action.

Contribution

It introduces new realizations of highest weight modules with respect to non-standard Gelfand-Tsetlin subalgebras and establishes conditions for their diagonalizable action.

Findings

Provided realizations of modules with non-standard Gelfand-Tsetlin subalgebras

Established sufficient conditions for diagonalizability

Enhanced understanding of module structures in Lie algebra representations

Abstract

In this paper we study realizations of highest weight modules for the complex Lie algebra $\mathfrak{gl}_n$ with respect to non-standard Gelfand-Tsetlin subalgebras. We also provide sufficient conditions for such subalgebras to have a diagonalizable action on these realizations.